We then have \(f\ast _G g\in L^p(G)\)  with
\(\Vert f\ast _G g\Vert _{L^p(G)}\le \Vert f\Vert _{L^1(G)}\Vert g\Vert _{L^p(G)}\) , if \(f\in L^1(G)\)  and \(g\in L^p(G)\) .
It is well known as a classical result in abstract harmonic analysis that the Banach function space \(L^1(G,\lambda _G)\)  is a Banach \(*\) -algebra with respect to the bilinear product \(\ast _G:L^1(G)\times L^1(G)\rightarrow L^1(G)\)  given by \((f,g)\mapsto f\ast _G g\) , with \(f\ast _G g\)  defined by (REF ) and involution given by \(f\mapsto f^{\ast _G}\)  where \(f^{*^G}(x):=\Delta _G(x^{-1})\overline{f(x^{-1})}\)  for \(x\in G\) . Also, for each \(p>1\) , the Banach function space \(L^p(G)\)  is a Banach left \(L^1(G)\) -module equipped with
the left module action \(\ast _G:L^1(G)\times L^p(G)\rightarrow L^p(G)\) 
given by \((f,g)\mapsto f\ast _G g\) , with \(f\ast g\)  is defined by (REF ), see [1]}, [2]}, [3]}, [4]}, [5]} and the classical list of references therein.
